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The human cerebral cortex has many bumps and grooves called gyri and sulci. Even though there is a high inter-individual consistency for the main cortical folds, this is not the case when we examine the exact shapes and details of the folding patterns. Because of this complexity, characterizing the cortical folding variability and relating them to subjects' behavioral characteristics or pathologies is still an open scientific problem. Classical approaches include labeling a few specific patterns, either manually or semi-automatically, based on geometric distances, but the recent availability of MRI image datasets of tens of thousands of subjects makes modern deep-learning techniques particularly attractive. Here, we build a self-supervised deep-learning model to detect folding patterns in the cingulate region. We train a contrastive self-supervised model (SimCLR) on both Human Connectome Project (1101 subjects) and UKBioBank (21070 subjects) datasets with topological-based augmentations on the cortical skeletons, which are topological objects that capture the shape of the folds. We explore several backbone architectures (convolutional network, DenseNet, and PointNet) for the SimCLR. For evaluation and testing, we perform a linear classification task on a database manually labeled for the presence of the "double-parallel" folding pattern in the cingulate region, which is related to schizophrenia characteristics. The best model, giving a test AUC of 0.76, is a convolutional network with 6 layers, a 10-dimensional latent space, a linear projection head, and using the branch-clipping augmentation. This is the first time that a self-supervised deep learning model has been applied to cortical skeletons on such a large dataset and quantitatively evaluated. We can now envisage the next step: applying it to other brain regions to detect other biomarkers.

We propose a local modification of the standard subdiffusion model by introducing the initial Fickian diffusion, which results in a multiscale diffusion model. The developed model resolves the incompatibility between the nonlocal operators in subdiffusion and the local initial conditions and thus eliminates the initial singularity of the solutions of the subdiffusion, while retaining its heavy tail behavior away from the initial time. The well-posedness of the model and high-order regularity estimates of its solutions are analyzed by resolvent estimates, based on which the numerical discretization and analysis are performed. Numerical experiments are carried out to substantiate the theoretical findings.

A popular method for variance reduction in observational causal inference is propensity-based trimming, the practice of removing units with extreme propensities from the sample. This practice has theoretical grounding when the data are homoscedastic and the propensity model is parametric (Yang and Ding, 2018; Crump et al. 2009), but in modern settings where heteroscedastic data are analyzed with non-parametric models, existing theory fails to support current practice. In this work, we address this challenge by developing new methods and theory for sample trimming. Our contributions are three-fold: first, we describe novel procedures for selecting which units to trim. Our procedures differ from previous work in that we trim not only units with small propensities, but also units with extreme conditional variances. Second, we give new theoretical guarantees for inference after trimming. In particular, we show how to perform inference on the trimmed subpopulation without requiring that our regressions converge at parametric rates. Instead, we make only fourth-root rate assumptions like those in the double machine learning literature. This result applies to conventional propensity-based trimming as well and thus may be of independent interest. Finally, we propose a bootstrap-based method for constructing simultaneously valid confidence intervals for multiple trimmed sub-populations, which are valuable for navigating the trade-off between sample size and variance reduction inherent in trimming. We validate our methods in simulation, on the 2007-2008 National Health and Nutrition Examination Survey, and on a semi-synthetic Medicare dataset and find promising results in all settings.

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. The relaxed micromorphic model is a generalized continuum that utilizes the $\Curl$ of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature...

We present new Neumann-Neumann algorithms based on a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, the Lagrange multiplier approach provides a coupled forward-backward optimality system, which can be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, nine variants can be found for the Neumann-Neumann algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.

Any interactive protocol between a pair of parties can be reliably simulated in the presence of noise with a multiplicative overhead on the number of rounds (Schulman 1996). The reciprocal of the best (least) overhead is called the interactive capacity of the noisy channel. In this work, we present lower bounds on the interactive capacity of the binary erasure channel. Our lower bound improves the best known bound due to Ben-Yishai et al. 2021 by roughly a factor of 1.75. The improvement is due to a tighter analysis of the correctness of the simulation protocol using error pattern analysis. More precisely, instead of using the well-known technique of bounding the least number of erasures needed to make the simulation fail, we identify and bound the probability of specific erasure patterns causing simulation failure. We remark that error pattern analysis can be useful in solving other problems involving stochastic noise, such as bounding the interactive capacity of different channels.

We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, where the law of the process is replaced by the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.

A new sparse semiparametric model is proposed, which incorporates the influence of two functional random variables in a scalar response in a flexible and interpretable manner. One of the functional covariates is included through a single-index structure, while the other is included linearly through the high-dimensional vector formed by its discretised observations. For this model, two new algorithms are presented for selecting relevant variables in the linear part and estimating the model. Both procedures utilise the functional origin of linear covariates. Finite sample experiments demonstrated the scope of application of both algorithms: the first method is a fast algorithm that provides a solution (without loss in predictive ability) for the significant computational time required by standard variable selection methods for estimating this model, and the second algorithm completes the set of relevant linear covariates provided by the first, thus improving its predictive efficiency. Some asymptotic results theoretically support both procedures. A real data application demonstrated the applicability of the presented methodology from a predictive perspective in terms of the interpretability of outputs and low computational cost.

In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large-scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms are also discussed in the numerical section.

One problem with researching cognitive modeling and reinforcement learning (RL) is that researchers spend too much time on setting up an appropriate computational framework for their experiments. Many open source implementations of current RL algorithms exist, but there is a lack of a modular suite of tools combining different robotic simulators and platforms, data visualization, hyperparameter optimization, and baseline experiments. To address this problem, we present Scilab-RL, a software framework for efficient research in cognitive modeling and reinforcement learning for robotic agents. The framework focuses on goal-conditioned reinforcement learning using Stable Baselines 3 and the OpenAI gym interface. It enables native possibilities for experiment visualizations and hyperparameter optimization. We describe how these features enable researchers to conduct experiments with minimal time effort, thus maximizing research output.